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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| Accès en ligne: | https://arxiv.org/abs/2604.15138 |
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| _version_ | 1866917414038405120 |
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| author | Charlesworth, Alison Ramsey, Christopher Strungaru, Nicolae |
| author_facet | Charlesworth, Alison Ramsey, Christopher Strungaru, Nicolae |
| contents | The 1-2-3 conjecture has been solved positively in 2024 for finite graphs and by extension for infinite graphs which are locally finite. The solution is non-constructive, and finding explicit solutions for large (or infinite) graphs is very hard. By exploiting the extra structure present in many non-periodic tilings, we find explicit solutions for the Chair (all three vertex placements), Non-Pinwheel, Pinwheel, Half-hex, Ammann-Beenker (two versions), Penrose Rhomb, and the Domino tilings. We prove that for any fully periodic tiling of the plane there exists a fully periodic solution, and provide an algorithm for finding such a solution. We give solutions for the fully periodic square, triangle and hexagonal lattices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_15138 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The 1-2-3 conjecture for polygonal tilings Charlesworth, Alison Ramsey, Christopher Strungaru, Nicolae Combinatorics The 1-2-3 conjecture has been solved positively in 2024 for finite graphs and by extension for infinite graphs which are locally finite. The solution is non-constructive, and finding explicit solutions for large (or infinite) graphs is very hard. By exploiting the extra structure present in many non-periodic tilings, we find explicit solutions for the Chair (all three vertex placements), Non-Pinwheel, Pinwheel, Half-hex, Ammann-Beenker (two versions), Penrose Rhomb, and the Domino tilings. We prove that for any fully periodic tiling of the plane there exists a fully periodic solution, and provide an algorithm for finding such a solution. We give solutions for the fully periodic square, triangle and hexagonal lattices. |
| title | The 1-2-3 conjecture for polygonal tilings |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.15138 |